Differentiation in Kinematics for JEE

Differentiation is a technique that is used to calculate the instantaneous rate of change of a function with respect to the independent variable. With the help of this method, a derivative of a function can be evaluated. In calculus, differentiation plays a major role in solving various complex problems in kinematics. In Mathematics, it is also used for determining and solving a mathematical equation. Differentiation is used for solving complex problems in the fields of Science, Engineering, and Mathematics. If there is one variable, say y and there is another independent variable x, then the differentiation y with respect to x is determined as \[\dfrac{dy}{dx}\].

Kinematics

Kinematics is the field of science that is used to analyse the motion of particles without considering the cause of their motion. For determining the motion of particles, various parameters are required to be defined which include displacement, velocity, and acceleration. According to calculus, it has been found that acceleration is the second derivative of displacement and the first derivative of velocity. Similarly, velocity is given as the first derivative of displacement. Examples of particle motion are a car moving on a straight road, a man walking on the street, a train moving from one station to another, etc.

Displacement

It is defined as the shortest distance between the initial point and the final point. Displacement is a vector quantity, which means that it has magnitude as well as direction. It can be positive, negative, or zero. The displacement of a particle can be written in terms of the function of the independent variable. For example, the displacement of a particle is given as a function of time which is shown below: 

\[s=4{{t}^{2}}-2t+3\]

In the above example, s is the displacement function of a particle with respect to time.

Velocity – Differentiation of Displacement

It is defined as the displacement travelled by the body per unit of time. It is a vector quantity which means that it has magnitude as well as direction. Velocity is different from speed as it is defined as the total distance travelled by the body per unit of time and it is a scalar quantity. The unit of velocity is metre/second \[(\dfrac{m}{s})\].

In calculus, velocity is determined as the rate of change of displacement per unit of time. In general terms, it can be said that velocity is the first derivative of displacement. Suppose, the displacement of a particle is a function of time, then the velocity of the particle will be determined by the differentiation of displacement of the particle. Velocity is the derivative of the displacement and it is written as

\[V=\dfrac{ds(t)}{dt}\]

For example, the displacement function of a particle is shown below. As velocity is the first derivative of displacement, it will help to evaluate velocity from the displacement function.

\[s(t)=4{{t}^{2}}-2t+3\]

\[V=\dfrac{ds(t)}{dt}\]

\[V=\dfrac{d(4{{t}^{2}}-2t+3)}{dt}\]

\[V=(2\times 4)t-2(1)\]

\[V=8t-2\]

Hence, the velocity of the particle is \[8t-2\]\[\dfrac{m}{s}\].

Acceleration – Differentiation of Velocity

It is defined as the change in velocity by the body per unit of time. It is a vector quantity which means that it has magnitude as well as direction. In calculus, acceleration is determined as the rate of change of velocity per unit of time. The unit of acceleration is \[\left(\dfrac{m}{{{s}^{2}}}\right)\]. 

In general terms, acceleration is said to be the second derivative of displacement and the first derivative of velocity. Differentiation of acceleration is called Jerk and can be understood as sudden rapid acceleration.  For the determination of the acceleration of a particle, the velocity function of the particle is differentiated with respect to time.  

Suppose, the velocity of a particle is a function of time and it is given as 

\[V(t)=8t-2\]

Then, the acceleration of the particle is determined as

\[A(t)=\dfrac{dV(t)}{dt}\]

\[A(t)=\dfrac{d(8t-2)}{dt}\]

\[A(t)=8\times 1\]

\[A(t)=8\]

Hence, the acceleration of the particle will be \[8\]\[\dfrac{m}{{{s}^{2}}}\].

Conclusion

This article concludes the basic information regarding the concept of differentiation and its application in the field of kinematics for the analysis of the motion of a particle. It explains the determination of all the required parameters like displacement, velocity, and acceleration with the help of differentiation. Differentiation is a technique of calculus that has been equipped for solving the complex problems of kinematics. 

In the given article, various conclusive results on displacement have been made as they can be positive, negative, and zero, and it can be written as a function of an independent variable i.e. time. Similarly, velocity has also been written as a function of time and can be evaluated by differentiating displacement with respect to time. Acceleration is also determined by differentiation velocity with respect to time when it is written as a function of time and we learnt that differentiation of acceleration is termed as Jerk. Examples have also been included for the proper understanding of this concept of kinematics.